DESCRIPTION (Adapted from applicant abstract): The objective of this project is to construct rigorous lower bounds on the convergence rates of Markov chains used in genetic analysis, in order that the Markov chain Monte Carlo (MCMC) technique can be used with greater confidence. This technique, since its introduction into segregation and linkage analysis in 1989, has had an ever increasing number of applications, and it is widely believed to be the ideal method for linkage analyses that take into account sources of uncertainty (such as allele frequencies, diagnostic misclassification, typing errors, map location uncertainities, epistasis, partial penetrance and sex effects), and also for segretation analyses that include multiple genes and multiple traits, interacting according to the complex patterns expected in neuropsychiatric disorders. It is also widely recognized that the major weakness of MCMC is the absence of any way to be certain that the Markov chain has adequately converged. While there are many "convergence diagnostics" in the literature, they can provide only necessary, not sufficient, conditions. This problem becomes 3specially worrisome when MCMC techniques are applied by general users for complex models with many parameters. Finding rigorous lower bounds on MCMC convergence rates will make possible secure guidelines for using the method, and lay the groundwork for more ambitious applications of MCMC in genetics in the future.